Bled Workshops in Physics
Vol. 4, No. 1 vTLv
Proceedings of the Mini-Workshop
Effective q-q Interaction (p. 95)
Bled, Slovenia, July 7-14, 2003
A simplified collective model of pion *
Borut T. Oblak
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, P.O. Box 2964, SI-1001 Ljubljana, Slovenia
Abstract. In order to test the accuracy of the aproximate methods commonly used for the Nambu - Jona-Lasinio model we study a simpler model which can be solved exactly. We find that the Random Phase Approximation gives reasonably good results if used in combination with the Hartree ground state (vacuum). On the other hand, the Tamm-Dancoff and Hermitian Operator Methods give even better results, but for the price of requiring a better approximation of the ground state.
1 Introduction
In the Nambu - Jona-Lasinio model (NJL), the vacuum properties and the pion excitation are usually calculated using the Hartree-Fock (HF) and Random Phase Approximations (RPA). We propose a simplied version of NJL which is appropriate to test the accuracy of these aproximate methods. The model preserves the main features of NJL and is simple enough to be solved exactly. For simplicity we limit ourselves to one flavour of quarks.
Since we shall deal with a finite number of quarks, it is convenient to start with the one-flavour NJL Hamiltonian written in the first-quantized form [1] and with a momentum cutoff A
N
H = Y_ (,5(k)h(k)p(k) + moP(k)) k=1
N N
- f Z ZOWPM + (iP^Tsik)) ■ (i(3il)T5fl))) ■
k=1 1=1
l=k
A A A A
■ H H H H 5pk+pi.Pk+Pipk, p0<Pk, Pii ■ pk pi pk pi
* Outline of the Diploma Thesis presented at the University of Ljubljana (Mentor: Mitja Rosina)
96 B. T. Oblak
2 The simple model
We made four approximations:
1.	We confined quarks in a finite volume V with periodic boundary conditions so that their momenta become discrete. Because the absolute values of momenta are limited, there is only a finite number of momenta avaliable. Therefore we have only a finite number 2N of single-particle states occupied by a finite number N of quarks.
2.	In the kinetic term of the Hamiltonian we take an average absolute value of momenta (P = |A) instead of their true values.
3.	The interaction changes only the quark's chirality and preserves its helicity, color and momentum which then label the quark. Therefore the quarks can be treated as distinguishable and Hartree is equivalent to Hartree-Fock.
4.	We include a quark selfinteraction so that the double sumations can be replaced by two single sumations. This contributes a trivial constant — gN.
The simplified Hamiltonian is:
N
^(,5 (k)h(k)P + mo =i
NN
■(X PM Z13 ^ + Z iPMT5(k) ^i
H =
k=1
N	N	N	N
We can introduce the operators:
K	1 -o	1
Jx = 2 P . h = 21|3ts '	= 275 '
which obey (quasi)spin commutation relations and allow us to make full use of the angular momentum algebra.
Also separate sums over quarks with right and left helicity
f 1+K(Tc).	f l-K(Tc).
k=1 k=1 as well as the total sum
+ Sa = ja(k)
k=1
obey the (quasi)spin commutation relations (a = x, y, z). The model Hamiltonian can then be written as
H = 2P(LZ — Sz) + 2moJ* — 2g(jX + ).
It commutes with L2 and S2 but not with Lz and Sz. Nevertheless, it is convenient to work in the basis |L, S, Lz, Sz).The Hamiltonian matrix elements can be
J
a
A simplified model of pion	97
easily calculated using the angular momentum algebra. By diagonalisation we then obtain the energy spectrum of the system.
The model has three model parameters: P, m0 and g. Instead of g we prefer to take G = g V/2 where V = i2N/A3 is the normalization volume since g decreases with increasing number of quarks while G stabilizes at large N.
We want to study the simple model in a physically interesting regime. Therefore we choose the three model parameters so that we fit three observable
1. We calculate the mass of dressed quark (M = 335 MeV) from the difference between the ground state energies (Eg) of the systems of N and N — 1 quarks. For the momentum of quark we take the average value (P) and we obtain
\/(E9(N) — Eg(N — 1))2 — P2 .
M = y (E
2. The mass of pion m2 should be 138 MeV. The pion corresponds to the first excited state of the system,
Ei = Ei — Eg => m„ = y E2 — p2 ,
where E1 is the energy of the first excited state and E2 is the pion energy. We determined the effective pion momentum p2 by the requirement, that the pion behaves as a Goldstone boson in the chiral limit and that p2 does not change much when the small quark mass term is introduced:
Pi = Ei (mo = 0) — Eg (mo = 0).
3. Instead of the pionic decay constant (f2 = 93 MeV) we prefer to fit the chiral condensate Q which is related to f2 through the Gell-Mann - Oakes - Renner relation
—Q = fimi/mo .
In this way we avoid the ambiguity how to introduce f2 in a one-flavour model, as well as the ambiguities with the effective momentum of the pion in
a finite volume. In our model, the chiral condensate is
1 N 2
Q = y<glX^l9> = ytelJxig). i=1
We compare the fitted values of model parameters for several values of N (Table 1). It is amusing that they are rather close to NJL parameters corresponding to two flavours and infinite number of quarks in the system [2].
3 Test of approximate methods - the vacuum
We compare the ground state (vacuum) energy Eg and the chiral condensate Q of the Hartree approximation with the exact solution.
The vacuum energy (Table 2) for N=48 and for the physically interesting value G = 40.1 MeVfm3 deviates only by 1.2%. The deviation decreases with
98 B. T. Oblak
Table 1. Model parameters (above) fitted to reproduce the observables (below).
N	12	24	36	48	NJL	exper.
G (MeV fmJ )	69.9	55.9	46.5	40.1	42.2	
mo (MeV)	26.0	15.9	11.8	9.6	5.5	
P (MeV)	484	557	613	659	473	
M (MeV)	335	335	335	335	335	335
m« (MeV)	138	138	138	138	138	138
fn (MeV)	93.0	93.0	93.0	93.0	93.0	93.0
Table 2. The energies Eg of the ground state for 48 quarks for P = 659 MeV and mo = 9.6 MeV and three values of G.
G (MeVfm3)	20.0	40.1	60.0
Exact Hartree	-32058.96 -31991.62	-32970.80 -32586.51	-37028.30 -36565.25
increasing N which hints that Hartre is a good large-N limit (we could not test it yet for large enough N). One should take some care, however, since in spite of the good agreement the Hartree ground state is still above the first (few) exact excited states in some of the studied cases.
Fig.1. Dependence of absolute value of the chiral condensate on the strength of interaction for 48 particles and P = 659 MeV. From above follow the lines for m0 = 9.6, 4.8, 2.4, 1.2, 0.6, 0.3 and 0 MeV. Exact (left) and Hartree (right) results are compared.
For a finite system we do not expect a sharp transition from the chirally symmetric to the chirally broken phase as a function of the interaction strength G. As a matter of fact, for m0 = 0 the system remains chirally symmetric, the order parameter Q remains zero. For a small but finite explicit symmetry breaking term m0 the system responds first with a small Q proportional to m0. For G larger than
A simplified model of pion
99
some critical value, however, Q starts to rise sharply (Fig.1). This is the analogue for spontaneous symmetry breaking in the case of a finite system. One expects a sharp phase transition if one makes the limit N —> oo faster than the limit mo —> 0.
On the other hand, one gets in the Hartree approximation a sharp phase transition already in the chiral limit mo = 0 and a slightly larger chiral condensate for mo > 0. This shows that the Hartree approximation strongly exaggerates the chiral symmetry breaking and in this way immitates the situation at N	even
at smaller N.
4 Test of approximate methods -1 and ct mesons.
The first excited state (negative parity) corresponds to pion and the second excited state (positive parity) corresponds to sigma meson. As approximate methods we study several particle-hole methods in which the ground state is excited by a one-body ("particle-hole") excitation operator.
In our case the low-lying states are symmetric under permutation of quark labels. Therefore the one-body excitation operators can be expressed as combination of quasispin operators Lx , Sx , iLy , iSy , Lz and Sz which we denote jointly by Bt, i = 1, ...6. Then we expand the excited states in the basis |i)
| exc) =£_ ct|i>, | i) = Bt|g.
i
The calculation is formulated in terms of a secular equation for the excitation energy ! and expansion coefficients ct
Different approximate methods differ in the proposition for the hamiltonian and overlap matrices
1.	In the Tamm-Dancoff method (TD) the basis | i) is taken literally and one obtains
Un = <j|(H - Eg)|i) = <g | Bj (H - Eg) Bt | g> and •Ajt = <j |i) = <g | Bj Bt|g).
2.	The Hermitian Operator Method (HOM) [3] is an approximation to TD which restricts the excitation operator to be either hermitian or antihermitian and relies on g being an exact ground state. This simplifies the evaluation of the matrix elements, but it makes a restriction to a smaller model space by decoupling the spaces generated by real hermitian (Lx , Sx , Lz and Sz) and real antihermitian (iLy and iSy) operators.
7-iji = ^{gl [Bj , [H, B|]] |g) and
j ^ <j|i> = <9 I Bj Bi | g)
100 B. T. Oblak
where the upper line in JVj i applies if Bt and Bj are both hermitian or both antihermitian and the lower line (0) otherwise.
3.	The Simple Operator Method (SOM) is even a more restrictive approximation to TD, it chooses only one of the listed one-body operators, iJy. Its succes in the description of the pion is based on the observation that such state is very close to the pionic excitation: <7r| iJv|g)/^/{g|j2|g) = 0.990 (for N = 48). It is even useful to calculate the two-pion excitation |2i) = —Jyy |g) — (g|-Jyy |g) |g).
4.	In the Random Phase Apoproximation one makes a risky but often sucessful assumption that there exists an excitation operator A = Y.t c-iBi whose adjoint kills the ground state
A | g = | exc), A | g = 0.
The inspiration comes from the creation and annihilation operators of the harmonic oscillator and it is a promissing approximation when one observes harmonic vibrational spectra. One then gets
j = <g |[Bj , [H.BÍ]]|g> and •Ají = <g |[Bj , Bi]| g).
5 Conclusion
We found that although the Hartree ground state energy differs from the exact ground state energy only by a small percentage for realistic model parameters, the energy difference between the Hartree and the exact ground state is comparable to the energy differences between the lowest exact excited states and the exact ground state. In spite of this, the Random Phase Approximation (RPA) gives a rather good approximation of the pion energy if used with the Hartree ground state (Table 3); as a matter of fact, it gives better results when used with the Hartree ground state than when used with the exact ground state. The condition A | g) = 0 for a one-body operator A is namely better fulfilled in the case of the Hartree ground state than in the case of the exact ground state. On the other hand, the Hermitian Operator Method (HOM), the Simple Operator Method (SOM) and the Tamm-Dancoff (TD) method fail for the Hartree ground state, but give very good results when used with the exact ground state.
Acknowledgement. Part of this work has been done at the Joief Stefan Institute within the Research Programme PO517 financed by the Ministry of Education, Science and Sport of Slovenia.
A simplified model of pion
101
Table 3. Exact excitation energies compared to several approximate methods for 48 quarks, P = 659 MeV and mo = 9.6 MeV and three values of G.
G (MeVfm3)		20.0	40.1	60.0
	state	!( MeV) parity	cu( MeV) parity	cu(MeV) parity
		exact solutions		
	ITT) l9>	1870.76 1.00 1870.76 -1.00 1848.51 1.00 916.91 1.00 916.46 -1.00 0.00 1.00	879.21 1.00 788.36 -1.00 788.33 1.00 365.20 1.00 319.65 -1.00 0.00 1.00	947.76 -1.00 647.98 1.00 579.88 -1.00 401.18 1.00 214.59 -1.00 0.00 1.00
		approximations of low-lying states computed from the exact ground state		
RPA	Itt)	917.06 1.00 916.59 -1.00	538.36 1.00 423.80 -1.00	1630.42 1.00 591.55 -1.00
ID	k> Itt) l9>	917.00 1.00 916.53 -1.00 0.48 1.00	423.54 1.00 337.51 -1.00 4.74 1.00	1365.81 1.00 246.99 -1.00 9.01 1.00
HOM	Itt) l9>	916.96 1.00 916.49 -1.00 0.48 1.00	413.93 1.00 333.98 -1.00 4.76 1.00	1223.81 1.00 243.37 -1.00 9.07 1.00
SOM	|2tt) Itt)	1859.35 1.00 916.49 -1.00	843.56 1.00 333.98 -1.00	609.14 1.00 243.37 -1.00
		approximations of low-lying states computed from the Hartree ground state		
RPA	k> Itt)	908.25 1.00 907.75 -1.00	656.92 1.00 260.35 -1.00	1691.41 1.00 229.05 -1.00
ID	k> Itt) l9>	976.01 1.00 975.67 -1.00 0.00 1.00	886.01 1.00 760.63 -1.00 0.00 1.00	1744.26 1.00 1083.67 -1.00 0.00 1.00
HOM	Itt) l9>	975.70 -0.01 773.19 0.55 0.00 1.00	763.78 0.11 584.56 0.40 0.00 1.00	1881.07 -0.34 1097.31 0.00 0.00 1.00
SOM	|2tt) Itt)	1965.87 1.00 975.67 -1.00	1540.17 1.00 760.63 -1.00	2156.63 1.00 1083.67 -1.00
References
1.	J. da Providencia, M. C. Ruivo and C. A. de Sousa, Phys. Rev. D 36,1882 (1987).
2.	M. Fiolhais, J. da Providencia, M. Rosina and C. A. de Sousa, Phys. Rev. C 56, 3311 (1997).
3.	M. Bouten, P. van Leuven, M. V. Mihailovic and M. Rosina, Nucl. Phys. A202, 127 (1973).